Chromatic Index of Signed Generalized Book Graphs and Signed Complete Graphs

Abstract

A signed graph $(G,σ)$ consists of a graph $G$ and the signature $σ: E(G) \rightarrow \{+1,-1\}$. An incidence of $G$ is a pair $(v,e)$, where $v$ is one of the end vertices of an edge $e \in E(G)$. A proper $q$-edge coloring $γ$ of signed graph $(G,σ)$ is an assignment of colors to incidences satisfying that $γ(v,e) = - σ(e) γ(w,e)$ for every edge $e=vw$ and for any two incidences $(v,e)$ and $(v,f)$, involving the same vertex, $γ(v,e) \neq γ(v,f)$. The chromatic index of a signed graph $(G,σ)$, denoted by $χ'(G,σ)$, is the minimum number $q$ for which $(G,σ)$ has a proper $q$-edge coloring. In this paper, we determine the chromatic index of signed generalized book graphs. We also determine the chromatic index of signed complete graphs of order up to six.

Figures (10)

• Figure 1: The generalized book graph $B(5,3,3)$.
• Figure 2: Two switching non-isomorphic signed generalized book graphs over $B(5,3,3)$. Throughout, dashed lines represent negative edges and solid lines represent positive edges.
• Figure 3: Proper edge colorings of all-positive graphs over $B(4,2,3)$ and $B(5,3,3)$.
• Figure 4: Proper edge colorings of some signed generalized book graphs.
• Figure 5: Proper edge colorings of some signed generalized book graphs with signature $\sigma_1$.

Theorems & Definitions (32)

• Definition 1
• Lemma 2.1
• Lemma 2.2
• Definition 2
• Lemma 2.3
• Lemma 2.4
• Theorem 2.5
• Theorem 2.6
• Theorem 2.7
• Proposition 2.8